[PDF] Assignment 7 - University of California Riverside

1 Closure Properties of Context-Free Languages

Context-free languages are not closed under intersection or complement. This will be shown later. 2. Page 3. 1.5 Intersection with a regular language.

Chapter 17: Context-Free Languages ?

Theorem: CFLs are not closed under complement. If L1 is a CFL then L1 may not be a CFL. Proof. They are closed under union. If they are closed under complement 

1 Closure Properties

Context-free languages are not closed under complementation. Proof. [Proof 1] Suppose CFLs were closed under complementation. Then for any two CFLs L1. L2

Languages That Are and Are Not Context-Free

Thus they are not closed under complementation. The Deterministic Context-Free Languages Are Closed Under Complement. Proof: Let L be a language such that L$ is 

Lecture Notes 12: Properties of Context-free Languages 1 Closure

- For a language L ? ?? L = ?? L. This shows that CFLs are not closed under set difference as well. Page 3. 1.2 Some 

CS 373: Theory of Computation

Let L1 and L2 be context free languages. L1 ? L2 is not necessarily context free! Proposition 2. CFLs are not closed under intersection. Proof.

CS154 slides

of CFL's is not closed under ?. difference is closed under intersection. ?Proof: L ? M = L – (L ... a CFL. ?Proof involves running a DFA in parallel.

Properties of Context-Free Languages

Proof: L ? M = L – (L – M). ?Thus if CFL's were closed under difference

Automaty a gramatiky - TIN071

27-Apr-2017 Context–free languages are closed under homomorphism. Proof: ... Theorem (CFL are not closed under difference nor complement).

Language-theoretic problems arising from Richelieu cryptosystems

Properties of Context-Free Languages - Stanford University

Nonclosure Under Difference We can prove something more general: Any class of languages that is closed under difference is closed under intersection Proof: L M = L – (L – M) Thus if CFL’s were closed under difference they would be closed under intersection but they are not

Closure Properties for Context-Free Languages

To show that the context-free languages are closed under union let AandBbe context-free lan-guages over an alphabet? and letGA=(VA?RASA)andGB=(VB?RBSB)be context-freegrammars that generateAandBrespectively By renaming the variables if necessary we assumethatVAis disjoint fromVB and that neither variable set contains the variableS

Assignment 7 - University of California Riverside

3) Show that the family of context-free languages is closed under reversal 4) Show that the family of context-free languages is not closed under difference in general but is closed under regular difference that is if L1 is context-free and L2 is regular then L1 – L2 is context-free

Are context-free languages closed?

Context-free languages are not closedunder intersection or complement. Thiswill be shown later. 1.5 Intersection with a regular language The intersection of a context-free language and a regular language is context-free (Theorem 3.5.2).

Is the intersection of a regular language and a context-free language?

Since A and B were arbitrary, we conclude that the intersection of a regular language and a context-free language is context-free. 10.4 Reverse We will now show that the context-free languages are closed under the operations reverse, pre?x, su?x, and substring.

What is the complement of a context-free language?

The complement of a context-free language can be context-free or not; the complement of a non-context free language can be context-free or not. Every regular language is context-free. Regular languages are closed under complement, so the complement of a regular language is regular.

Do two context-free grammars generate the same language?

Different context-free grammars can generate the same context-free language. It is important to distinguish the properties of the language (intrinsic properties) from the properties of a particular grammar (extrinsic properties). The language equality question (do two given context-free grammars generate the same language?) is undecidable .